You can work at UPS while doing general aviation and CFI work. Be cautious of fatigue though, you can't really throw boxes all night and then safely fly all day...if anything happened they would violate you.

But you'll need excellent turbine credentials, typically from military aviation or ACMI cargo airlines to have even a remote chance at UPS. At an absolute minimum 1000 hours as a regional CA, but more likely you'll need international experience on heavy aircraft (ie widebodies or big military transports). I've heard current employees might get preferential interviews for UPS pilot slots, but you probably won't be able to stay employed at UPS while working at a regional.